Hardy Littlewood Sobolev Inequality Research Articles

Fractional Musielak spaces: study of nonlocal elliptic problem with Choquard-logarithmic nonlinearity

In this study, we thoroughly examine a specific type of complex mathematical problems involving a fractional ψ κ , y ( ⋅ ) -Laplacian operator and a Choquard-logarithmic nonlinearity, combined with a real parameter, known as non-homogeneous elliptic problems. Our approach involves setting-specific conditions for the Choquard nonlinearities and the continuous function ψ κ , y . This allows us to successfully identify multiple solutions to these problems. Our analysis is conducted in the area of fractional Musielak spaces. We rely heavily on the mountain pass theorem and Ekeland's variational principle, as well as the Hardy–Littlewood–Sobolev inequality, which is essential for supporting the theoretical basis of our research.

Stability of Hardy-Littlewood-Sobolev inequalities with explicit lower bounds

In this paper, we establish the stability for the Hardy-Littlewood-Sobolev (HLS) inequalities with explicit lower bounds. By establishing the relation between the stability of HLS inequalities and the stability of fractional Sobolev inequalities, we also give the stability of the higher and fractional order Sobolev inequalities with the lower bounds. This extends to some extent the stability of the first order Sobolev inequalities with the explicit lower bounds established by Dolbeault, Esteban, Figalli, Frank and Loss in [18] to the higher and fractional order case. Our proofs are based on the competing symmetries, the continuous Steiner symmetrization inequality for the HLS integral and the dual stability theory. As another application of the stability of the HLS inequality, we also establish the stability of Beckner's [4] restrictive Sobolev inequalities of fractional order s with 0<s<n2 on the flat sub-manifold Rn−1 and the sphere Sn−1 with the explicit lower bound. When s=1, this implies the explicit lower bound for the stability of Escobar's first order Sobolev trace inequality [19] which has remained unknown in the literature.

Functional Inequalities on Symmetric Spaces of Noncompact Type and Applications

The aim of this paper is to begin a systematic study of functional inequalities on symmetric spaces of noncompact type of higher rank. Our first main goal of this study is to establish the Stein–Weiss inequality, also known as a weighted Hardy–Littlewood–Sobolev inequality, for the Riesz potential on symmetric spaces of noncompact type. This is achieved by performing delicate estimates of ground spherical function with the use of polyhedral distance on symmetric spaces and by combining the integral Hardy inequality developed by Ruzhansky and Verma with the sharp Bessel-Green-Riesz kernel estimates on symmetric spaces of noncompact type obtained by Anker and Ji. As a consequence of the Stein–Weiss inequality, we deduce Hardy–Sobolev, Hardy–Littlewood–Sobolev, Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities on symmetric spaces of noncompact type. The second main purpose of this paper is to show the applications of aforementioned inequalities for studying nonlinear PDEs on symmetric spaces. Specifically, we show that the Gagliardo-Nirenberg inequality can be used to establish small data global existence results for the semilinear wave equations with damping and mass terms for the Laplace–Beltrami operator on symmetric spaces.

Fractional integration and optimal estimates for elliptic systems

In this paper we give an affirmative answer to the Euclidean analogue of a question of Bourgain and Brezis concerning the optimal Lorentz estimate for a Div–Curl system: If F∈L1(R3;R3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F \\in L^1(\\mathbb {R}^3;\\mathbb {R}^3)$$\\end{document} satisfies divF=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {div}F=0$$\\end{document} in the sense of distributions, then the function Z=curl(-Δ)-1F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Z=\ ext {curl} (-\\Delta )^{-1} F$$\\end{document} satisfies curlZ=FdivZ=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ ext {curl } Z&= F \\\\ \ ext {div } Z&= 0 \\end{aligned}$$\\end{document}and there exists a constant C>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C>0$$\\end{document} such that ‖Z‖L3/2,1(R3;R3)≤C‖F‖L1(R3;R3).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Vert Z\\Vert _{L^{3/2,1}(\\mathbb {R}^3;\\mathbb {R}^3)} \\le C\\Vert F\\Vert _{L^{1}(\\mathbb {R}^3;\\mathbb {R}^3)}. \\end{aligned}$$\\end{document}Our proof relies on a new endpoint Hardy–Littlewood–Sobolev inequality for divergence free measures which we obtain via a result of independent interest, an atomic decomposition of such objects.

Critical mass phenomenon for a parabolic–elliptic multispecies chemotaxis system in a two‐dimensional disk

A parabolic–elliptic Keller–Segel model for multipopulations in a two‐dimensional unit ball will be considered in this paper. For the initial boundary value problem with mutually attractive populations, the global existence of bounded solutions has been proved through the logarithmic Hardy–Littlewood–Sobolev inequality for system when the initial masses satisfy the subcritical condition. Moreover, based on the moments technique, there exists a blow‐up solution when the initial masses satisfy the supercritical case. In a word, a critical mass phenomenon has been established in this multispecies chemotaxis system.

On quantum Sobolev inequalities

We investigate the quantum analogue of the classical Sobolev inequalities in the phase space, with the quantum Sobolev norms defined in terms of Schatten norms of commutators. These inequalities provide an uncertainty principle for the Wigner–Yanase skew information, and also lead to new bounds on the Schatten norms of the Weyl quantization in terms of its symbol. As an intermediate tool, we obtain the analogue of Hardy–Littlewood–Sobolev's inequalities for a semiclassical analogue of the convolution, and introduce quantum Besov spaces. Explicit estimates are obtained on the optimal constants.

Improved Hardy–Littlewood–Sobolev Inequality on $${\mathbb{S}^n}$$ under Constraints

Improved Hardy–Littlewood–Sobolev Inequality on $${\mathbb{S}^n}$$ under Constraints

Existence results for a nonlinear integral system of the Hardy–Hénon type in a bounded domain

This paper investigates the existence and nonexistence of positive solutions for a weighted integral system associated with the weighted Hardy–Littlewood–Sobolev inequalities in a bounded domain Ω⊂Rn. We apply the mountain pass theorem to demonstrate the existence of a solution to this system and prove the nonexistence of a solution for it by means of the Pohozaev identity. We also use the above methods to prove the existence and nonexistence of positive solutions for a Hénon-type integral system.

Stability of Hardy Littlewood Sobolev inequality under bubbling

In this note we will generalize the results deduced in Figalli and Glaudo (Arch Ration Mech Anal 237(1):201–258, 2020) and Deng et al. (Sharp quantitative estimates of Struwe’s Decomposition. Preprint http://arxiv.org/abs/2103.15360, 2021) to fractional Sobolev spaces. In particular we will show that for sin (0,1), n>2s and nu in mathbb {N} there exists constants delta = delta (n,s,nu )>0 and C=C(n,s,nu )>0 such that for any function uin dot{H}^s(mathbb {R}^n) satisfying,u-∑i=1νU~iH˙s≤δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\| u-\\sum _{i=1}^{\ u } \ ilde{U}_{i}\\right\\| _{\\dot{H}^s} \\le \\delta \\end{aligned}$$\\end{document}where tilde{U}_{1}, tilde{U}_{2},ldots tilde{U}_{nu } is a delta -interacting family of Talenti bubbles, there exists a family of Talenti bubbles U_{1}, U_{2},ldots U_{nu } such that u-∑i=1νUiH˙s≤CΓif2s<n<6s,Γ|logΓ|12ifn=6s,Γp2ifn>6s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\| u-\\sum _{i=1}^{\ u } U_{i}\\right\\| _{\\dot{H}^s} \\le C\\left\\{ \\begin{array}{ll} \\Gamma &{} \ ext{ if } 2s< n < 6s,\\\\ \\Gamma |\\log \\Gamma |^{\\frac{1}{2}} &{} \ ext{ if } n=6s, \\\\ \\Gamma ^{\\frac{p}{2}} &{} \ ext{ if } n > 6s \\end{array}\\right. \\end{aligned}$$\\end{document}for Gamma =left| Delta u+u|u|^{p-1}right| _{H^{-s}} and p=2^*-1=frac{n+2s}{n-2s}.

Local uniqueness of blow-up solutions for critical Hartree equations in bounded domain

In this paper we are interested in the following critical Hartree equation -Δu=(∫Ωu2μ∗(ξ)|x-ξ|μdξ)u2μ∗-1+εu,inΩ,u=0,on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} -\\Delta u =\\displaystyle {\\Big (\\int _{\\Omega }\\frac{u^{2_{\\mu }^*} (\\xi )}{|x-\\xi |^{\\mu }}d\\xi \\Big )u^{2_{\\mu }^*-1}}+\\varepsilon u,~~~\ ext {in}~\\Omega ,\\\\ u=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\ ext {on}~\\partial \\Omega , \\end{array}\\right. } \\end{aligned}$$\\end{document}where Nge 4, 0<mu le 4, varepsilon >0 is a small parameter, Omega is a bounded domain in mathbb {R}^N, and 2_{mu }^*=frac{2N-mu }{N-2} is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. By establishing various versions of local Pohozaev identities and applying blow-up analysis, we first investigate the location of the blow-up points for single bubbling solutions to above the Hartree equation. Next we prove the local uniqueness of the blow-up solutions that concentrates at the non-degenerate critical point of the Robin function for varepsilon small.

Regularity results for Choquard equations involving fractional p‐Laplacian

AbstractIn this paper, first we address the regularity of weak solution for a class of p‐fractional Choquard equations: where is a smooth bounded domain, and such that , and is a continuous function with at most critical growth condition (in the sense of the Hardy–Littlewood–Sobolev inequality) and F is its primitive. Next, for , we discuss the Sobolev versus Hölder minimizers of the energy functional J associated with the above problem, and using that we establish the existence of the local minimizer of J in the fractional Sobolev space . Moreover, we discuss the aforementioned results by adding a local perturbation term (at most critical in the sense of Sobolev inequality) in the right‐hand side in the above equation.

Existence of ground state solutions for a Choquard double phase problem

In this paper we study quasilinear elliptic equations driven by the double phase operator involving a Choquard term of the form −Lp,qa(u)+|u|p−2u+a(x)|u|q−2u=∫RNF(y,u)|x−y|μdyf(x,u)inRN,where Lp,qa is the double phase operator given by Lp,qa(u)≔div(|∇u|p−2∇u+a(x)|∇u|q−2∇u),u∈W1,H(RN),0<μ<N, 1<p<N, p<q<p+αpN, 0≤a(⋅)∈C0,α(RN) with α∈(0,1] and f:RN×R→R is a continuous function that satisfies a subcritical growth. Based on the Hardy–Littlewood–Sobolev inequality, the Nehari manifold and variational tools, we prove the existence of ground state solutions of such problems under different assumptions on the data.

Semi-classical states for the Choquard equations with doubly critical exponents: Existence, multiplicity and concentration

In this paper, we are concerned with a class of Choquard equation with the lower and upper critical exponents in the sense of the Hardy–Littlewood–Sobolev inequality. We emphasize that nonlinearities with doubly critical exponents are totally different from the well-known Berestycki–Lions-type ones. Working in a variational setting, we prove the existence, multiplicity and concentration of positive solutions for such equations when the potential satisfies some suitable conditions. We show that the number of positive solutions depends on the profile of the potential and that each solution concentrates around its corresponding global minimum point of the potential in the semi-classical limit.

Existence of ground‐state solutions for p‐Choquard equations with singular potential and doubly critical exponents

AbstractWe are concerned with the following Choquard equation: where , , , , is the p‐Laplacian, is the Riesz potential, and F is the primitive of f which is of critical growth due to the Hardy–Littlewood–Sobolev inequality. Under different range of θ and almost necessary conditions on the nonlinearity f in the spirit of Berestycki–Lions‐type conditions, we divide this paper into three parts. By applying the refined Sobolev inequality with Morrey norm and the generalized version of the Lions‐type theorem, some existence results are established. It is worth noting that our method is not involving the concentration–compactness principle.

Multiple solutions for a fractional Choquard problem with slightly subcritical exponents on bounded domains

This paper is devoted to study a fractional Choquard problem with slightly subcritical exponents on bounded domains. When the exponent of the convolution type nonlinearity tends to the fractional critical one in the sense of Hardy–Littlewood–Sobolev inequality, we obtain the existence of multiple positive solutions via Lusternik–Schnirelmann category and nonlocal global compactness. Moreover, we prove that the topology of the domain furnishes a lower bound for the number of positive solutions.

Sharp reversed Hardy–Littlewood–Sobolev inequality with extension kernel

In this paper, we prove the following reversed Hardy–Littlewood–Sobolev inequality with extension kernel: $$\int _{\mathbb R_+^n}\int _{\partial\mathbb R^n_+}\frac{x_n^\beta}{|x-y|^{n-\alpha}}f(y)g(x)\,dy\,dx\geq C_{n,\alpha ,\beta ,p}\|f\|_{L^{p}(\partia

Hardy–Littlewood–Sobolev inequality and existence of the extremal functions with extended kernel

In this paper, we consider the following Hardy–Littlewood–Sobolev inequality with extended kernel (0.1)\begin{equation} \int_{\mathbb{R}_+^{n}}\int_{\partial\mathbb{R}^{n}_+} \frac{x_n^{\beta}}{|x-y|^{n-\alpha}}f(y)g(x) {\rm d}y{\rm d}x\leq C_{n,\alpha,\beta,p}\|f\|_{L^{p}(\partial\mathbb{R}_+^{n})} \|g\|_{L^{q'}(\mathbb{R}_+^{n})}, \end{equation}for any nonnegative functions $f\in L^{p}(\partial \mathbb {R}_+^{n})$, $g\in L^{q'}(\mathbb {R}_+^{n})$ and $p,\,\ q'\in (1,\,\infty )$, $\beta \geq 0$, $\alpha +\beta &gt;1$ such that $\frac {n-1}{n}\frac {1}{p}+\frac {1}{q'}-\frac {\alpha +\beta -1}{n}=1$.We prove the existence of all extremal functions for (0.1). We show that if $f$ and $g$ are extremal functions for (0.1) then both of $f$ and $g$ are radially decreasing. Moreover, we apply the regularity lifting method to obtain the smoothness of extremal functions. Finally, we derive the sufficient and necessary condition of the existence of any nonnegative nontrivial solutions for the Euler–Lagrange equations by using Pohozaev identity.

Critical curve for a two-species chemotaxis model with two chemicals in * *Supported by National Natural Science Foundation of China (12171218), LiaoNing Revitalization Talents Program (XLYC2007022) and Key Project of Education Department of Liaoning Province (LJKZ0083).

This paper deals with a two-species chemotaxis model with two chemicals in . In our previous work (2019 Nonlinearity 32 4762–78), the critical mass was obtained that the solutions exist globally if m 1 m 2 − 4π(m 1 + m 2) < 0, and the finite time blow-up of solutions may occur if m 1 m 2 − 4π(m 1 + m 2) > 0, where m 1 and m 2 describe the initial mass of the two species, respectively. In the present paper we furthermore determine that the critical situation belongs to the global existence case, namely, the system admits global solutions if m 1 m 2 − 4π(m 1 + m 2) = 0. To apply the key logarithmic Hardy–Littlewood–Sobolev inequality of vector form for the critical case, we should establish a positive lower bound for L 1-norms of the two species in exterior domains uniformly for t > 0.

Sign-changing solutions to the critical Choquard equation

We consider the following Choquard problem −Δu=λu+f(x)(|x|−μ∗|u|2μ∗)|u|2μ∗−2uinΩ,u=0on∂Ω, where 0<λ<λ1(Ω),|Ω|<∞. 2μ∗ is the upper critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. We prove the existence of sign-changing solution by combining Nehari manifold method with the Ljusternik–Schnirelman theory. The main results extend and complement the earlier works (Guo et al., 2019; Moroz and Schaftingen, 2015).

On a stochastic Hardy–Littlewood–Sobolev inequality with application to Strichartz estimates for a noisy dispersion

In this paper, we investigate a stochastic Hardy–Littlewood–Sobolev inequality. Due to the non-homogenous nature of the potential in the inequality, we show that a constant proportional to the length of the interval appears on the right-hand-side. As a direct application, we derive local Strichartz estimates for randomly modulated dispersions and solve the Cauchy problem of the critical nonlinear Schrödinger equation.